| Project/Area Number |
16540193
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka Kyoiku University |
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Principal Investigator |
KATAYAMA Yoshikazu Osaka Kyoiku Univ., Division of Math.Science, Professor, 教育学部, 教授 (10093395)
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| Co-Investigator(Kenkyū-buntansha) |
FUJII Masatoshi Osaka Kyoiku Univ., Dept.of Math., Professor, 教育学部, 教授 (10030462)
KAWAKAMI Satoshi Nara University of Education, Dept.of Math., Professor, 教育学部, 教授 (20161284)
OUCHI Motoo Osaka prefecture univ., Dept.of Math., Professor, 大学院・理学系研究科, 教授 (70127885)
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| Project Period (FY) |
2004 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | outer action / 3-cohomology / outer conjugate / AFD因子環 / 3次のコサイクル / G-核 / 外部自己同型 / 従順な離散群 / III型因子環 / コホモロジー群 / ハイゼンベルグ群 |
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| Research Abstract |
Let N be a normal subgroup of a group G and its quotient group Q=G/N. In the former work, the definition of modular obstruction was given by the use of continuous crossed product. According to a discrete decomposition of AFD III_λ-factor (by A.Connes), we may consider the group Q_m={(p,t)∈Q×R : mod α_p=t mod T'Z}. Consequently the extended modular obstruction on G is isomorphic to a certain Torus valued 3-cohomology on Q_m and the characteristic invariant on G is also so. Then we have the extended HJR-exact sequence for AFD-factor of type III_λ. This exact sequence is the first column in the following diagram which shows the relation between it and an ordinary HJR-exact sequence :
【numerical fomula】
We proved the above diagram, by giving induction and reduction theory for groupoids and their dimension shifting theorem, On the other hand, this Torus valued extended modular invariant is a complete invariant, up to outer conjugacy, for outer actions on AFD-factor of type III_λ and this discrete extended HJR-exact sequence help us to give a model of outer action with a given modular obstruction.
Let A be an element of SL(2 Z). We consider a crossed product group G(A)=Z^2×_A Z by A. We parameterize the 3-cohomology group H^3(G(A),T)〓T. We give explicitly the model outer action with some condition associated with t in H^3 (G(A),T) and we also give the way how to get the element t from a given outer action. By cotistructing a model action with a given invariant, we have to compute HJR-exact sequence for the group G(A) explicitly, for example, 2-cohomology group characteristic invariant for the group G(G).
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